L9n20

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L9n20's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_11,17,12,16 X_15,9,16,18 X_17,13,18,12 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -8, -2, 9}, {8, -1, -3, 4}, {-9, 2, -5, 7, -4, 3, -6, 5, -7, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$u 3 + v u 2 -2 v w u 2 + w u 2 -2 u 2 - v u + 2 v w u - w u + 2 u - v w$ (db)
Jones polynomial	$- q 8 + 2 q 7 -4 q 6 + 5 q 5 -4 q 4 + 6 q 3 -3 q 2 + 3 q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-2 z 4 a -4 + 3 z 2 a -2 -7 z 2 a -4 + 3 z 2 a -6 + 5 a -2 -10 a -4 + 6 a -6 - a -8 + 2 a -2 z -2 -5 a -4 z -2 + 4 a -6 z -2 - a -8 z -2$ (db)
Kauffman polynomial	$z 7 a -5 + z 7 a -7 + 4 z 6 a -4 + 6 z 6 a -6 + 2 z 6 a -8 + 3 z 5 a -3 + 5 z 5 a -5 + 3 z 5 a -7 + z 5 a -9 -11 z 4 a -4 -16 z 4 a -6 -5 z 4 a -8 -6 z 3 a -3 -18 z 3 a -5 -15 z 3 a -7 -3 z 3 a -9 + 6 z 2 a -2 + 18 z 2 a -4 + 15 z 2 a -6 + 3 z 2 a -8 + 11 z a -3 + 21 z a -5 + 13 z a -7 + 3 z a -9 -7 a -2 -14 a -4 -10 a -6 -2 a -8 -5 a -3 z -1 -9 a -5 z -1 -5 a -7 z -1 - a -9 z -1 + 2 a -2 z -2 + 5 a -4 z -2 + 4 a -6 z -2 + a -8 z -2$ (db)

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

2 is the signature of L9n20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = 0$	${\mathbb Z}^{3}$	${\mathbb Z}^{3}$
$r = 1$	${\mathbb Z}^{3}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$